Thread: Derivative of a constant complex function

Let be an analytic function on a disc D.

Is it true that:

is constant

with ' denoting the derivative wrt to complex variable z. Or do we have to include some conditions on the partial derivatives wrt x,y (where Re(z)=x and Im(z)=y) ?

Thanks for anyhelp!

Hey Ant.

This is true and you can use the fact that all analytic functions in a region are infinitely differentiable with the caveat that differentiating a constant gives 0.

Thanks.

Do we also have the following:

both partial derivatives with respect to and are also zero.


(I think perhaps the above is false and only the direction holds??)


Thanks!

Yes we should have and you could use the Cauchy-Riemann equations to show this.